3.94 \(\int \frac{(d+e x^2) (a+b \text{sech}^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=238 \[ -\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\frac{8 b c^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x}+\frac{4 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (30 c^2 d+49 e\right )}{1225 x^5}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{49 x^7} \]

[Out]

(b*d*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(49*x^7) + (b*(30*c^2*d + 49*e)*Sqrt[(1 + c*x)^(-1)
]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(1225*x^5) + (4*b*c^2*(30*c^2*d + 49*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*
Sqrt[1 - c^2*x^2])/(3675*x^3) + (8*b*c^4*(30*c^2*d + 49*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2
])/(3675*x) - (d*(a + b*ArcSech[c*x]))/(7*x^7) - (e*(a + b*ArcSech[c*x]))/(5*x^5)

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Rubi [A]  time = 0.121574, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 6301, 12, 453, 271, 264} \[ -\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\frac{8 b c^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x}+\frac{4 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (30 c^2 d+49 e\right )}{1225 x^5}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{49 x^7} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^8,x]

[Out]

(b*d*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(49*x^7) + (b*(30*c^2*d + 49*e)*Sqrt[(1 + c*x)^(-1)
]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(1225*x^5) + (4*b*c^2*(30*c^2*d + 49*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*
Sqrt[1 - c^2*x^2])/(3675*x^3) + (8*b*c^4*(30*c^2*d + 49*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2
])/(3675*x) - (d*(a + b*ArcSech[c*x]))/(7*x^7) - (e*(a + b*ArcSech[c*x]))/(5*x^5)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6301

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] &&
((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ
[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-5 d-7 e x^2}{35 x^8 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\frac{1}{35} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-5 d-7 e x^2}{x^8 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\frac{1}{245} \left (b \left (-30 c^2 d-49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^6 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{b \left (30 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{1225 x^5}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\frac{\left (4 b c^2 \left (-30 c^2 d-49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^4 \sqrt{1-c^2 x^2}} \, dx}{1225}\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{b \left (30 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{1225 x^5}+\frac{4 b c^2 \left (30 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3675 x^3}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}+\frac{\left (8 b c^4 \left (-30 c^2 d-49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx}{3675}\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{b \left (30 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{1225 x^5}+\frac{4 b c^2 \left (30 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3675 x^3}+\frac{8 b c^4 \left (30 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3675 x}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.200744, size = 117, normalized size = 0.49 \[ \frac{-105 a \left (5 d+7 e x^2\right )+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (15 d \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )+49 e x^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )\right )-105 b \text{sech}^{-1}(c x) \left (5 d+7 e x^2\right )}{3675 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + b*ArcSech[c*x]))/x^8,x]

[Out]

(-105*a*(5*d + 7*e*x^2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(
5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d + 7*e*x^2)*ArcSech[c*x])/(3675*x^7)

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Maple [A]  time = 0.191, size = 160, normalized size = 0.7 \begin{align*}{c}^{7} \left ({\frac{a}{{c}^{2}} \left ( -{\frac{d}{7\,{c}^{5}{x}^{7}}}-{\frac{e}{5\,{c}^{5}{x}^{5}}} \right ) }+{\frac{b}{{c}^{2}} \left ( -{\frac{{\rm arcsech} \left (cx\right )d}{7\,{c}^{5}{x}^{7}}}-{\frac{{\rm arcsech} \left (cx\right )e}{5\,{c}^{5}{x}^{5}}}+{\frac{240\,{c}^{8}d{x}^{6}+392\,{c}^{6}e{x}^{6}+120\,{c}^{6}d{x}^{4}+196\,{c}^{4}e{x}^{4}+90\,{c}^{4}d{x}^{2}+147\,{c}^{2}{x}^{2}e+75\,{c}^{2}d}{3675\,{c}^{6}{x}^{6}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arcsech(c*x))/x^8,x)

[Out]

c^7*(a/c^2*(-1/7/c^5*d/x^7-1/5*e/c^5/x^5)+b/c^2*(-1/7*arcsech(c*x)/c^5*d/x^7-1/5*arcsech(c*x)*e/c^5/x^5+1/3675
*(-(c*x-1)/c/x)^(1/2)/c^6/x^6*((c*x+1)/c/x)^(1/2)*(240*c^8*d*x^6+392*c^6*e*x^6+120*c^6*d*x^4+196*c^4*e*x^4+90*
c^4*d*x^2+147*c^2*e*x^2+75*c^2*d)))

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Maxima [A]  time = 1.02732, size = 223, normalized size = 0.94 \begin{align*} \frac{1}{245} \, b d{\left (\frac{5 \, c^{8}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{7}{2}} + 21 \, c^{8}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} + 35 \, c^{8}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 35 \, c^{8} \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c} - \frac{35 \, \operatorname{arsech}\left (c x\right )}{x^{7}}\right )} + \frac{1}{75} \, b e{\left (\frac{3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} + 10 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 15 \, c^{6} \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c} - \frac{15 \, \operatorname{arsech}\left (c x\right )}{x^{5}}\right )} - \frac{a e}{5 \, x^{5}} - \frac{a d}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arcsech(c*x))/x^8,x, algorithm="maxima")

[Out]

1/245*b*d*((5*c^8*(1/(c^2*x^2) - 1)^(7/2) + 21*c^8*(1/(c^2*x^2) - 1)^(5/2) + 35*c^8*(1/(c^2*x^2) - 1)^(3/2) +
35*c^8*sqrt(1/(c^2*x^2) - 1))/c - 35*arcsech(c*x)/x^7) + 1/75*b*e*((3*c^6*(1/(c^2*x^2) - 1)^(5/2) + 10*c^6*(1/
(c^2*x^2) - 1)^(3/2) + 15*c^6*sqrt(1/(c^2*x^2) - 1))/c - 15*arcsech(c*x)/x^5) - 1/5*a*e/x^5 - 1/7*a*d/x^7

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Fricas [A]  time = 1.95465, size = 352, normalized size = 1.48 \begin{align*} -\frac{735 \, a e x^{2} + 525 \, a d + 105 \,{\left (7 \, b e x^{2} + 5 \, b d\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (8 \,{\left (30 \, b c^{7} d + 49 \, b c^{5} e\right )} x^{7} + 4 \,{\left (30 \, b c^{5} d + 49 \, b c^{3} e\right )} x^{5} + 75 \, b c d x + 3 \,{\left (30 \, b c^{3} d + 49 \, b c e\right )} x^{3}\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{3675 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arcsech(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/3675*(735*a*e*x^2 + 525*a*d + 105*(7*b*e*x^2 + 5*b*d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) -
(8*(30*b*c^7*d + 49*b*c^5*e)*x^7 + 4*(30*b*c^5*d + 49*b*c^3*e)*x^5 + 75*b*c*d*x + 3*(30*b*c^3*d + 49*b*c*e)*x
^3)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{8}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*asech(c*x))/x**8,x)

[Out]

Integral((a + b*asech(c*x))*(d + e*x**2)/x**8, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arcsech(c*x))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arcsech(c*x) + a)/x^8, x)